A decomposition theorem for χ1-convex sets

A set S in $\text{R}$ is said to be χ-convex if and only if S does not contain a visually independent subset having cardinality χ. It is natural to ask when an χ-convex set may be expressed as a countable union of convex sets. Here it is proved that if S is a closed χ-convex set in the plane and $\text{R}$ has at most finitely many bounded components, then S is a countable union of convex sets. A parallel result holds in $\text{R}$ when S is a closed χ-convex set which contains all triangular regions whose relative boundaries are in S. However, the result fails for arbitrary χ-convex sets, even in the plane.     

Visible shorelines inR d

Summary LetC be a closed set inR ^d and letj be a fixed integer,j 1. The setS R ^d ~C is said to have aj-partition relative toC if there existj or fewer pointsc ₁,, c _j ofC such that each point ofS sees via the complement ofC at least one pointc _i. For every triple of integersd, p, j withd 0, p d + 1, j 1, there exists a smallest integerf(d, p, j) such that the following is true: IfC is a convexd-polytope inR ^d havingp vertices and ifS R ^d ~C, S has aj-partition relative toC if and only if everyf(d, p, j)-member subset of S has such a partition.ForC a convex polytope inR ² andS R ² ~C, all points ofS see via the complement ofC a common neighborhood in the boundary ofC if and only if every three points ofS see via the complement ofC such a neighborhood.A weak analogue of this result holds for arbitrary compact convex sets inR ^d .     

Semi-empirical molecular orbital calculations on some tautomers of cytosine

The stability of a series of cytosine tautomers has been investigated using both all valence electron and -electron SCF methods. The effect of general and selective scaling of the bond lengths was studied in the CNDO/2 calculations. The enol tautomer of cytosine exceeded the keto form in stability in these calculations. -electron SCF CI calculations were performed on the cytosine tautomers using two different parameterization schemes. The calculated transitions obtained for the enol form showed a close proximity to that of the keto and imino structures. Comparison was made with the experimental electronic spectrum in water and acetonitrile. No clear distiction between the keto and enol forms could be made on the basis of these calculations.

---

Zusammenfassung Die StabilitÄt einer Reihe von Cytosin-Tautomeren wurde mit einer Allvalenzelektronen- und mit einer -Elektronen-SCF-Methode untersucht. Die Auswirkung einer allgemeinen als auch selektiven Skalierung der BindungslÄngen wurde bei den CNDO2-Berechnungen untersucht. Das Enoltautomere des Cytosins übertrifft nach diesen Rechnungen die Ketoform an StabilitÄt. Bei den -Elektronen-SCF-CI-Rechnungen wurden zwei Methoden der Parametrisierung angewandt. Die für die Enolform berechneten übergÄnge zeigten eine gro\e Ähnlichkeit zu denjenigen der Keto- und Iminostrukturen. Die berechneten Werte wurden mit dem experimentellen Spektrum von Cytosin in Wasser sowie Acetonitril verglichen. Auf Grund der Berechnungen konnte keine klare Unterscheidung zwischen der Keto- und der Enolform getroffen werden.

---

Résumé Etude de la stabilité d'une série de tautomères de la cytosine par les méthodes SCF à électrons de valence et à électrons . L'effet d'ajustement global ou selectif des longueurs de liaison a été étudié dans le cadre CNDO/2. Le tautomère énolique de la cytosine s'avère dans ces calculs plus stable que la forme cétonique. Des calculs d'électrons SCF CI ont été effectués sur les tautomères de la cytosine en utilisant deux schémas différents pour la paramétrisation. Les transitions calculées obtenues pour la forme énolique montrent une grande analogie avec celles des structures cétoniques et iminiques. La comparaison a été faite avec le spectre électronique expérimental dans l'eau et l'acétonitrile. Aucune distinction claire entre les formes cétoniques et énoliques n'a pu Être faite à partir de ces calculs.

     

Automatic poisson peak harvesting for high throughput protein identification

High throughput identification of proteins by peptide mass fingerprinting requires an efficient means of picking peaks from mass spectra. Here, we report the development of a peak harvester to automatically pick monoisotopic peaks from spectra generated on matrix-assisted laser desorption/ionisation time of flight (MALDI-TOF) mass spectrometers. The peak harvester uses advanced mathematical morphology and watershed algorithms to first process spectra to stick representations. Subsequently, Poisson modelling is applied to determine which peak in an isotopically resolved group represents the monoisotopic mass of a peptide. We illustrate the features of the peak harvester with mass spectra of standard peptides, digests of gel-separated bovine serum albumin, and with Escherictia coli proteins prepared by two-dimensional polyacrylamide gel electrophoresis. In all cases, the peak harvester proved effective in its ability to pick similar monoisotopic peaks as an experienced human operator, and also proved effective in the identification of monoisotopic masses in cases where isotopic distributions of peptides were overlapping. The peak harvester can be operated in an interactive mode, or can be completely automated and linked through to peptide mass fingerprinting protein identification tools to achieve high throughput automated protein identification.     

Intersecting families of convex cover order m

Let S=cl(int S) be a subset of the plane, Q the set of points of local non-convexity of S, Qfinite, with p a point in (bdry S ker S) Q. If denotes the collection of allmaximal families of wedges of S having convex cover order m, m1, then the members of are characterized in the following manner: For W _o a wedge of S, W _o is in if and only if for every w _o in W _o and every collection of points w ₁, ...w _m in S, then [w _i, w_j] S for at least one pair i, j, 0i<jm.AMS(MOS) Subject Classification 1970: Primary 52A10, 52A40.     

Clear visibility,starshaped sets,and finitely starlike sets

Let S be an arbitrary nonempty set in R^d. The following results are true for every k, 0kd: the dimension of ker S is at least k if and only if every countable family of boundary points of S is clearly visible from a common k-dimensional neighborhood in S. Similarly, ker S contains a k-dimensional -neighborhood if and only if every countable family of boundary points of S is clearly visible from a common k-dimensional -neighborhood in S.In the plane, we have the following results concerning finitely starlike sets: for S an arbitrary nonempty set in R², S is finitely starlike if every three points of cl S are clearly visible from a common point of S. In case S –R² and int cl SS=, then S is finitely starlike if and only if every three points of S are visible from a common point of S. In each case, the number 3 is best possible.     

Liquid chromatography determination of residue levels on apples treated with cetylpyridinium chloride

Cetylpyridinium chloride (CPC) has been found to be effective in reducing microbial contamination in apples. A sensitive and specific HPLC method was developed to determine CPC residues in apples treated with CPC. This method involves ion exchange solid-phase extraction, and the use of stearylpyridinium chloride (SPC) as internal standard. Limit of quantitation, was 0.5 microg/ml of CPC for the apple ethanolic extracts. The observed residues in apple (2.35-4.35 microg/g of apple) were lower than those previously reported for chicken and beef. The method is specific, sensitive, reproducible and accurate.     

Large-order estimates for ground-state energy perturbation series

A general treatment is given, using path integral methods, of obtaining accurate estimates on the rate of growth at large order of the perturbation coefficients for the lowest eigenvalue (ground-state energy) of a large class of anharmonic oscillators. Simple sufficient conditions are given on the potentialV(x) so that accurate upper and lower bounds on the perturbation coefficients may be derived. Several examples are given which generalize previous results. Examples from Euclidean quantum field theory are also considered.This paper is dedicated to the memory of Mark Kac.